Duality for p-adic differential equations (attention jour inhabituel)

P-adic differential equations are an important tool in studying the geometry and arithmetic of algebraic varieties over a finite field, or more generally a field of positive characteristic. I will give a briefs urvey of recent work of Poineau and Pulita on the finiteness of the de Rham cohomology of a connection on a Berkovich curve, and then describe a version of the Lefschetz duality theorem such connections in the non-solvable case. The latter is joint work with Andrea Pulita.